Stochastic Differential Geometry, a Coupling Property, and Harmonic Maps

A Riemannian manifold has the Brownian coupling property if two Brownian motions can be constructed on it, with arbitrary initial points, and such that they are sure to meet at some time. While Euclidean space has this property, simply‐connected manifolds with negative curvature bounded away from zero do not. Such a coupling is then not possible even if Γ‐martingales of bounded dilatation are used rather than Brownian motions. A generalised little Picard theorem for harmonic maps is proved using these results.